Pullback Map

A pullback is a general categorical operation appearing in a number of mathematical contexts, sometimes going under a different name. If T:V->W is a linear transformation between vector spaces, then T^*:W^*->V^* (usually called transpose map or dual map because its associated matrix is the transpose of T) is an example of a pullback map.

In the case of a diffeomorphism and smooth manifold, a very explicit definition can be formulated. Given an r-form alpha on a manifold M_2, define the r-form T^*(alpha) on M_1 by its action on an r-tuple of tangent vectors (X_1,...,X_r) as the number T^*(alpha)(X_1,...,X_r)=alpha(TX_1,...,TX_r). This defines a map on r-forms and is the pullback map.

See also

Category, Pushforward Map

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Cite this as:

Weisstein, Eric W. "Pullback Map." From MathWorld--A Wolfram Web Resource.

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